Did our first antiderivative problem by thinking carefully about the power rule working in reverse. Took apart the notation/symbols for indefinite integration (a synonym of antiderivatives or antidifferentiation) and saw that the solution to such a problem was a family of functions rather than 1 particular function or 1 particular number. Formalized the reverse power rule through a few examples and a formula and added it along with many other straightforward antiderivative rules to our formulas booklet (straightforward because they mimic the derivative rules from last semester).
Worked through several examples, including ones with radicals and requiring rewriting into negative exponents. Looked at 2 tricky cases, one yielding ln(x) rather than using the reverse power rule and another involving e^x. Looked at the derivative/antiderivative relationship in abstraction with a cow-milk-cheese analogy,
Started a brief intro to finding particular solutions to a simple differential equation by "finding C" by plugging given x and y values into the result of an antiderivative and solving for C.
Notes from board
Homework
p. 251 #7-26, 50
remember to check calcchat.com for odd solutions + help
Resources
rationale behind reverse power rule: link
many great integral examples link
Warm up was an antiderivative problem to solve with a given point to produce a particular solution (also known as 'finding C'). Went over HW, Then did several chain rule problems mentally to observe patterns, especially that the derivative of the part on the inside (the unchanged part) is multiplying the outer function's derivative. So evidence of the chain rule having been applied is to have the relationship where one part of an integrand is the derivative of another part. We saw this through several examples, all of which involved 'tweaking' or 'cheating' by multiplying by a constant to turn the almost-chain into what it was we wanted. We accommodated for this cheat or tweak by multiplying the outside of the integral by the reciprocal, so that the 2 changes we make have the net effect of multiplying by zero.
Notes from board
Homework
any 8 from the front, and any 8 from the back of this handout
Resources
reverse chain rule concept video example video
some more examples (made by meee!) link